Subgroup ($H$) information
| Description: | $C_2^{10}.C_2^7:S_4$ |
| Order: | \(3145728\)\(\medspace = 2^{20} \cdot 3 \) |
| Index: | $1$ |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Generators: |
$\langle(11,13)(12,14)(19,21)(20,22), (3,5)(4,6)(15,17)(16,18), (3,19)(4,20)(5,21) \!\cdots\! \rangle$
|
| Derived length: | $5$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a semidirect factor, nonabelian, and a Hall subgroup. Whether it is a direct factor or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{10}.C_2^7:S_4$ |
| Order: | \(3145728\)\(\medspace = 2^{20} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(201326592\)\(\medspace = 2^{26} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | Group of order \(201326592\)\(\medspace = 2^{26} \cdot 3 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^{10}.C_2^7:S_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |